Mathematics

# A Mathematical Model for Evaluation of Assets Returns in a Volatile Economy

Contents

## ABSTRACT

In recent years, advance in technology have made it possible for the stock markets to trade in real time and also for large dataset to be available for statistical analysis.

Thus, we examined the impact of macroeconomic variables on the stock returns of 114 companies listed on the Nigerian Stock Exchange Market. We have established the mathematical framework required to solve our model and perform various empirical analysis on the stock market data and macroeconomic variable. The formulated Macroeconomic Factor models are deployed to evaluate the effects of the macroeconomic variables on a volatile economy and Ordinary Least Square procedure are deployed to estimate the parameters of the model. We apply the model to the available data and discovered that the stock market return volatility is influenced by the selected macroeconomic variables; Gross Domestic Product, Inflation, Foreign Exchange Rate, Unemployment, Interest Rate, Price of Crude Oil and Money supply.

Approval i
Certification ii
Dedication iii
Acknowledgments iv
Abstract v
1 Introduction 1
1.1 Background of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Hypothesis of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Aims and Objectives of the Study . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Scope and Limitations of the Study . . . . . . . . . . . . . . . . . . . . 6
1.6 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Literature Review 10
3 Method 21
3.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Assets and Returns . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Assets Pricing Models . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.3 Macroeconomic Variables Description . . . . . . . . . . . . . . . 28
3.2 Research Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Formulation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Existing Model by Izedonmi et al (2011) . . . . . . . . . . . . . 37
3.4 Macroeconomic Factor Model for Returns . . . . . . . . . . . . . . . . 37
3.5 Unit Root Test for the Stock Data and the Macroeconomic Variables . 39
3.5.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.2 Non-Stationarity and Stationarity Test . . . . . . . . . . . . . . 40
3.6 Data and Data Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Results and Discussion 44
4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Descriptive Statistics of the Stock Data . . . . . . . . . . . . . . 44
4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Discussion of the Hedge Ratio (i0) . . . . . . . . . . . . . . . . 47
4.2.2 Effects of Inflation (i 1t) . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Effects of Exchange Rate (i 2t) . . . . . . . . . . . . . . . . . . 48
4.2.4 Effects of Gross Domestic Product (i 3t) . . . . . . . . . . . . . 48
4.2.5 Effects of Unemployment Rate (i 4t) . . . . . . . . . . . . . . . 48
4.2.6 Effects of Interest Rate (i5t) . . . . . . . . . . . . . . . . . . . 48
4.2.7 Effects of Price of Crude Oil (i 6t) . . . . . . . . . . . . . . . . 49
4.2.8 Effects of Money Supply, M2 (i 7t) . . . . . . . . . . . . . . . . 49
5 Summary, Conclusion and Recommendation 50
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Appendix A 52
Appendix B 109
Appendix C 117

## 1.1 Background of the Study

Nigeria is supposed to be one of the world’s richest nation if the natural reserve were used as a measure of wealth. In reality, oil reserve have been a curse to Nigeria instead of a blessing as the Agricultural, solid minerals and other non-oil sectors which were the main source of export have been neglected. This has made crude oil to be the main product of Nigerian export, thus giving an unfavorable balance of trade and payment which greatly affects the Nigerian economy in particular exchange rate, unemployment rate and inflation.

In recent years Nigeria has become a crude oil exporting and finished products (from crude oil) importing country, the balance of payment for the import and export may either be favourable or unfavourable. This volatility of oil prices have varying consequences on Nigeria as will reap the benefit of high oil prices as well as experience the unfavorable terms of trade in our external sector that can be transferred into the economy in the long run from oil imports. This situation makes the value of the exchange rate to increase thereby reducing the purchasing power of local currency and increasing the cost of importing raw materials and production. This change in exchange rate makes trade and investment more challenging since the decision makers cannot predict the approximate value of the exchange rate, thus resulting to inflation as the prices of most commodities will rise rapidly with little or no incentive to search for cheaper substitute that could help keep the production cost low.

The uncertainty about the future value of Naira and real interest rates makes businessmen or decision makers less willing to take risk in investing in long term projects.

As inflation rises businessmen shift their investments to gold and real estate instead of holding of stocks and bonds. The interest rate volatility results to less business investment and a reduction in the real economic growth as the supply of funds available for businessmen to borrow reduces thereby raising the cost of investment and maybe eventually the closure of the business. This is evidenced in the increase in interest rate to account for inflationary uncertainty, thus the cost of borrowing by businessmen and consumers increases substantially.

Investors always react to changes in market news so as to minimize the risk of investing in order to maximize product. Thus they tend to select the stock to invest in based on looking at the historic data of the adjusted closing price of the various stocks and trying to predict the future outcome. This has increased the interest of financial analyst, policy makers and academics in financial markets and the quest to forecast their performance with the increasing impact of macroeconomic variables. However, in Nigeria there is less attention in the study of the relationship between stock prices and fundamental economic activities.

The relationship between macroeconomic variables and stock market has been discussed in previous research papers such as Chen et al (1986), Abraham (2010) that deal with this topic in various ways. It is expected that the macroeconomic factors should affect the prices of stocks which in turn should influence the real economy. During the last two decades, empirical studies relating to this subject matter began to be a topic of interest with new improved methods to analyze the correlation between scale and real economic world. The major problem is distinguishing the magnitude in terms of size and direction of the relationship that exist between some macroeconomic variables and stock market. The results achieved from previous research by Chen et al (1986), Gay (2008), Zhu (2012) among others which will be discussed more in literature review show no significant change over time, though the evidence remains relevant for new markets trying to enter global financial world. They considered risk and returns as the two factors that can determine the value of financial assets. For emerging economy, these two factors are not enough determinant as to whether or not to invest in a particular asset. Markowitz (1952) stated the golden rule underlining the theory of investment that investors seek either to maximize returns at a given level of risk or to minimize risk at a given level of returns on their investment.

Generally speaking, Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) are the two broad frameworks for pricing financial assets. In particular, the CAPM is a one factor model which relies mostly on a measure, beta, which stresses the sensitivity of asset unpredictability to the unpredictability of the entire market.

On the other hand, the APT is a multi-factor model, which traces the expected return on a number of securities to their sensitivities. The Nigerian Stock Exchange (NSE) was established in 1960 as the Lagos Stock Exchange. As of December 31, 2013, it has about 200 listed companies with a total market capitalization of about N12.88 trillion (80.8 billion USD). The NSE is regulated by the Securities and Exchange Commission (SEC), which has the mandate of Surveillance over the stock exchange market in order to forestall violation of market rules and to deter and detect unfair manipulations and trading practices.

Data on listed companies performances are published daily, weekly, monthly, quarterly and annually. The Nigerian Stock Exchange has been operating an Automated Trading System (ATS) since April 27; 1999; with dealers trading through a network of computers connected to a server with facility for remote trading and surveillance.

Consequently, many stock brokers trade on-line from their offices in Lagos and from all the thirteen branches across the country. The Exchange is in the process of establishing more branches for on-line real time trading. Trading on the Exchange starts at 9.30 a.m. every business day and closes at 2.30 p.m.

In order to encourage direct foreign investment into Nigeria, the government have to review the legislation preventing the ow of foreign capital into the country. This has allowed foreign brokers to enlist as dealers on the Nigerian Stock Exchange, and investors of any nationality are free to invest. Nigerian companies are also allowed multiple and cross border listings on foreign markets. It is expected that an effective and efficient market should re ect the real economy; that is to say, that the stock market prices should incorporate the effect of changes in the macroeconomic variables such as inflation, GDP, exchange rate among others. This thesis does not use NSE all share prices index as the measure of stock market movements as different sectors and/or companies within the same sector respond differently to changes in the macroeconomic variables. Several research work mostly in the developed countries deal with recent economic development and problems, especially about the relationship between returns and macroeconomic variables. However, stock market interactions with macroeconomic variables remain a topic of interest to the developing countries. As the market is not perfectly integrated with global financial world, local macroeconomic factors can influence the stock market returns more than global risk factors. In the light of previous empirical evidence this research work purposes to model empirically the effect of macroeconomic variables and the selected stock data of 114 companies from different sectors of the economy using micro-foundations. According to Jerey(2012), the advantage of building macroeconomic models from micro-foundations is that we can begin to understand the relationship between macroeconomic outcomes and the individual decisions of the agents in the economy.

## 1.2 Statement of the Problem

The rate of fluctuation in the returns on stocks and the continuous decrease and/or increase in several macroeconomic variables has created panic in the Nigeria Stock Market. Studies on the interactions among the selected macroeconomic variables and selected stock market returns in Nigeria Stock Exchange market is the main purpose of this research. The presence of correlations between the stock price and macroeconomic variables can be explained, conversely, if the results support significance of macroeconomic variables and stock market returns, we can use them as a price determination tool for share prices. Most researchers have focused on systematic risk such as consumer price index, inflation, interest rate and so on as sources of risk. This means that in the long run, the returns on an asset should re ect the true situation of the economy. Many studies carried out in the developed world such as the US, the UK among others have attempted to establish the relationship between assets returns and economic indicators (Chen et al (1986), Gay (2008), Zhu (2012)).

The importance of establishing these relationships to investors is extremely imperative given that the risk faced by investors may be traced to the changing values of these economic variables. Several empirical studies have tested these relationships using the CAPM and APT on stock data of various countries and their results show that the theory has the possibility of explaining the returns on such capital markets. In developing and emerging capital markets of Africa including that of Nigeria, there are no widespread studies relating stock markets returns with macroeconomic variables such as interest rates, inflation, price of crude oil and money supply among others which to a large extent are expected to affect stock market activities. We seek to nd out if macroeconomic variables have any effect on the Nigeria stock exchange (NSE) returns?

We shall use the factor model which is based on the arbitrage pricing theory (APT) framework to examine the relationship between selected macroeconomic variables and the Nigerian stock market.

## 1.3 Hypothesis of the Study

Based on general intuitive theory suggested by Fama (1981) and previous literatures on this topic, we hypothesis that there is no relationship between the selected macroeconomic variables such as in ation, money supply, consumer price index, exchange rate, price of crude oil, gross domestic product and the returns of the 114 companies.
Mathematically,

0 = 1 = 2 = 3 = 4 = 5 = 6 = 7 = 0
Where
0 is the effect of hedge ratio (a measure for liquidity),
1 is the effect of in ation rate,
2 is the effect of exchange rate,
3 is the effect of gross domestic product,
4 is the effect of unemployment rate,
5 is the effect of interest rate,
6 is the effect of price of crude oil,
7 is the effect of money supply.

## 1.4 Aims and Objectives of the Study

### 1.4.1 Aims

The aim of this study is to critically investigate the macroeconomic factor model in order to identify the effects of macroeconomic variables on the performance and efficiency of Nigerian Stock Exchange Market in a Volatile economic situation.

### 1.4.2 Objectives

The objectives of this study are to formulate a model for the macroeconomic variables, use macroeconomic factor model to analyze the effects of the macroeconomic variables on the 114 companies listed on the Nigerian stock exchange market and determine the liquidity of the Nigerian stock exchange.

## 1.5 Scope and Limitations of the Study

The specific goal of this work is to determine the effect of selected macroeconomic variables on the historic data of asset returns of 114 rms listed on the Nigeria stock exchange (NSE). The results from this research may not give an accurate replication of what goes on in all other markets because the features of two different markets or group of stock markets differs hence their reaction to the selected macroeconomic variables will differ. The macroeconomic variables selected namely; Money supply, Exchange rate, Gross Domestic Product, Inflation, interest rate, unemployment rate and crude oil prices are factors that we have applied to the macroeconomic factor model. We have used the R statistical programming language to analyze the financial data and in estimating the macroeconomic factor model.

## 1.6 Motivation

We saw that previous models on the effect of macroeconomic variables and stock returns such as in studies by Chen et al (1986), Maysami et al(2004), Kuwornu et al (2011), Izedonmi et al (2011) among others, were based on either the all share index or the sectoral index and used a selection of less than six (6) macroeconomic factors. However, we discovered through research that the macroeconomic variables are the determinant of the state of an economy. For instance, a close investigation of the developed country shows that unemployment is inversely proportional to inflation because the developed economy are efficient and they engage all available resources including human skills in production using the principle of comparative advantage. Nigeria has assured socio- political and economic setbacks which has affected the quality of the macroeconomic policy and thus tends not to follow most of these economic principles as postulated by the developed countries.

In light of the above, we see that the existing models may not have properly taken care of the situation of addressing the effect of macroeconomic variables on the stock returns bearing in mind the volatility of the Nigerian economy. Hence, our desire to remodel the multifactor model taking into account more macroeconomic variables and liquidity. More so, our model build bridges between micro-level structures and macro-level relationships among variables.

## 1.7 Main Contributions

We combined the macroeconomic variables and the individual stock returns in order to achieve a better understanding of how the Nigerian economy behaves. Also, we considered the volatility of the macroeconomic variables and liquidity. This model builds bridges between the underlying micro-level structures and macro-level relationships among the variables.

## 1.8 Glossary

Stock: A share of a company held by an individual or group.

Asset: refers to a resource with economic value that an individual, corporation, state or country owns or controls with the expectation that it will provide future benefit.

Return: The gain or loss of an asset in a particular period.

Volatility: is a measure for the variation of price of a financial asset over time.

Macroeconomics: the study of the economy as a whole, and the variables that control the macro-economy.

Portfolio: A grouping of financial assets such as stocks, bonds and cash equivalents among others held by an investor.

Risk: means uncertainty in future returns from an investment.

Idiosyncratic risk: risk that is associated with an individual asset.

Exports: The Amount of Domestically Produced Goods Sold on Foreign Soil.

Imports: The Amount of Goods Produced on Foreign Soil Purchased Domestically.

Option: In nance, an option is a contract which gives the buyer (the owner) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date.

NYSE: New York Stock Exchange.

AMEX: American Express Financial Services Company

NASDAQ: National Association of Securities Dealers Automated Quotations

SENSEX: is an abbreviation of the Bombay Exchange Sensitive Index

FMCG: Fast-moving consumer goods

FII: Foreign Institutional Investment

CMR: Call Money Rate

DF: Dickey Fuller

AR(p): Autoregressive Process of order p, p=1,2, . . .

MA(q): The q Order Moving Average

ISE: Istanbul Stock Exchange

APT: Arbitrage Pricing Theory

CAPM: Capital Assets Pricing Model